====== Simple 2-Species ======

\( \require{newcommand} \)
\( \newcommand{\diff}[2]{\frac{d{#1}}{d{#2}}} \)

Following the [[..:Simple 1-State:start]] analysis we continue by considering the simple 2-species model.

===== Species =====

We distinguish a //species// from a //state// by the criterion that it is possible for an individual to switch //states// but not //species//.  So the concept of //species// is a subset of //state//.  For convenience we will restrict ourselves to two species (denoted by \(X\) and \(Y\)) with one state, each.

===== Reactions =====

Since each species has only one state, each has [[..:Simple 1-State:start#Reactions | five possible intraspecific reactions]]:
\[
\begin{array}{rclrcll}
  \emptyset & \xrightarrow{\alpha_x} & X,         & \emptyset & \xrightarrow{\alpha_y} & Y         & \text{(spontaneous creation)} \\
  X   & \xrightarrow{\beta_x}        & 2 X,       & Y   & \xrightarrow{\beta_y}        & 2 Y       & \text{(induced creation)} \\
  2 X & \xrightarrow{\gamma_x}       & 3 X,       & 2 Y & \xrightarrow{\gamma_y}       & 3 Y       & \text{(pair-induced creation)} \\
  X   & \xrightarrow{\delta_x}       & \emptyset, & Y   & \xrightarrow{\delta_y}       & \emptyset & \text{(spontaneous removal)} \\
  2 X & \xrightarrow{\epsilon_x}     & X,         & 2 Y & \xrightarrow{\epsilon_y}     & Y         & \text{(induced removal)}.
\end{array}
\]

==== Interspecific reactions ====

Each species could also influence the other, as well.  Such interspecific reactions require the presence of an individual of each species in the reaction set so there are only four possible [[..:Simple 1-State:start#Simplicity | simple]] reactions:
\[
\begin{array}{cll}
  X + Y & \xrightarrow{\nu_x} & 2 X + Y   & \text{(benefit to }X\text{)} \\
  X + Y & \xrightarrow{\nu_y} & X + 2 Y   & \text{(benefit to }Y\text{)} \\
  X + Y & \xrightarrow{\mu_x} & Y         & \text{(detriment to }X\text{)} \\
  X + Y & \xrightarrow{\mu_y} & X         & \text{(detriment to }Y\text{).}
\end{array}
\]

Depending on the relative magnitude of the rate constants, all six possible categories of [[wp>biological interaction]] are represented, as follows:
^  \(\nu_x - \mu_x\)  ^  \(\nu_y - \mu_y\)  ^ Type of interaction                                                              ^
|          0          |          0          | [[wp>biological interaction#Neutralism| Neutralism]]                             |
|         --          |          0          | [[wp>biological interaction#Amensalism| Amensalism]]                             |
|          +          |          0          | [[wp>Commensalism]]                                                              |
|         --          |         --          | [[wp>Competition (biology)| Competition]]                                        |
|          +          |          +          | [[wp>Mutualism (biology)| Mutualism]]                                            |
|          +          |         --          | [[wp>biologyical interaction#Antagonism| Antagonism]] (Predation or Parasitism)  |



Together, these 14 reactions fully describe all possible two-species simple systems.

===== Dynamics =====

The population densities of the two species, \(x\) and \(y\), will behave as
\[
\begin{array}{rcrrlrll}
  \diff{x}{t} & = & \alpha_x & + \beta'_x & x & + \gamma'_x & x^2 & + \nu'_x x\, y \\
  \diff{y}{t} & = & \alpha_y & + \beta'_y & y & + \gamma'_y & y^2 & + \nu'_y x\, y
\end{array}
\]
where
\[
\begin{array}{rll}
  \beta'  & \equiv \beta - \delta    & \text{(excess induced creation)} \\
  \gamma' & \equiv \gamma - \epsilon & \text{(excess pair-induced creation)} \\
  \nu'    & \equiv \nu - \mu         & \text{(excess interspecific creation)}.
\end{array}
\]

==== Sans immigration ====

The general solution to the dynamics is difficult but the special case without immigration (\(\alpha_x=\alpha_y=0\)) is well understood as the [[wp>competitive Lotka–Volterra equations#Two_species | two-species competitive Lotka-Volterra equations]]:
\[
\begin{array}{rcl}
  \diff{x}{t} & = & r_x x \left( 1 - \frac{x + A_x y}{K_x} \right) \\
  \diff{y}{t} & = & r_y y \left( 1 - \frac{y + A_y x}{K_y} \right)
\end{array}
\]
where the parameter mapping is
\[
\begin{array}{rcl}
  r_z & = & \beta'_z \\
  K_z & = & \beta'_z / (-\gamma'_z) \\
  A_z & = & \nu'_z / \gamma'_z.
\end{array}
\]

==== Conic sections ====

The [[wp>matrix representation of conic sections]] helps to categorize the possible dynamics in the general case.  The general form of the polynomial rates for the matrix representation is
\[
\begin{array}{rcrrrrrl}
  \diff{x}{t} & = & A_x x^2 & + B_x x y & + C_x y^2 & + D_x x & + E_x y & + F_x \\
  \diff{y}{t} & = & A_y x^2 & + B_y x y & + C_y y^2 & + D_y x & + E_y y & + F_y.
\end{array}
\]
In our case this means \(C_x=E_x=A_y=D_y=0\).  

=== Matrix ===

As a consequence, the \(A_Q\) matrix representations can only be degenerate if
\[
\begin{array}{lcl}
  |A_{Q,x}|=0 & \text{iff} & B_x^2 F_x = 0 \\
  |A_{Q,y}|=0 & \text{iff} & B_y^2 F_y = 0.
\end{array}
\]
We will use the degeneracy conditions to classify the nullclines.

=== Minor matrix ===

Computing the minor matrices,
\[
  A_{11} = \left( 
    \begin{array}{cc}
      A   & B/2 \\
      B/2 & C
    \end{array}
  \right)
\]
further allows to classify the nullclines.  
\[
\begin{array}{lcl}
  |A_{11,x}|<0 & \text{iff} & B_x^2 > 0 \\
  |A_{11,x}|=0 & \text{iff} & B_x^2 = 0 \\
  |A_{11,y}|<0 & \text{iff} & B_y^2 > 0 \\
  |A_{11,y}|=0 & \text{iff} & B_y^2 = 0
\end{array}
\]
since \(|A_{11}|=A C - B^2/4\) and either \(A=0\) or \(C=0\).

=== Nullclines ===

The matrix and minor matrix gives the following information about each respective nullcline:

^       Condition        ^ Nullcline(s)                              ^
|        \(B=0\)         | two (possibly coincident) parallel lines  |
|    \(B\neq 0, F=0\)    | two intersecting lines                    |
|  \(B\neq 0, F\neq 0\)  | hyperbola                                 |



